71. Consider a non-homogeneous system of linear equations representing mathematically an overdetermined system. Such a system will be
72. The dimension of the null space of the matrix \[\left[ {\begin{array}{*{20}{c}}
0&1&1 \\
1&{ - 1}&0 \\
{ - 1}&0&1
\end{array}} \right]\] is
73. X and Y are non-zero square matrices of size n × n. If XY = 0n × n then
74. A is m × n full rank matrix with m > n and $$I$$ is an identity matrix. Let matrix A' = (ATA)-1AT, Then, which one of the following statement is TRUE?
75. The minimum and the maximum eigen values of the matrix \[\left[ {\begin{array}{*{20}{c}}
1&1&3 \\
1&5&1 \\
3&1&1
\end{array}} \right]\] are -2 and 6, respectively. What is the other eigen value?
76. Consider the matrix \[\left[ {\begin{array}{*{20}{c}}
5&{ - 1} \\
4&1
\end{array}} \right]\] . Which one of the following statements is TRUE for the eigen values and eigen vectors of this matrix?
77. Let M be a real 4 × 4 matrix. Consider the following statements:
S1: M has 4 linearly independent eigenvectors.
S2: M has 4 distinct eigenvalues.
S3: M is non-singular (invertible).
Which one among the following is TRUE?
S1: M has 4 linearly independent eigenvectors.
S2: M has 4 distinct eigenvalues.
S3: M is non-singular (invertible).
Which one among the following is TRUE?
78. The smallest and largest Eigen values of the following matrix are \[\left[ {\begin{array}{*{20}{c}}
3&{ - 2}&2 \\
4&{ - 4}&6 \\
2&{ - 3}&5
\end{array}} \right]\]
79. Eigen values of a matrix \[{\text{S}} = \left[ {\begin{array}{*{20}{c}}
3&2 \\
2&3
\end{array}} \right]\] are 5 and 1. What are the eigen values of the matrix S2 = SS?
80. If any two columns of a determinant \[{\text{P}} = \left| {\begin{array}{*{20}{c}}
4&7&8 \\
3&1&5 \\
9&6&2
\end{array}} \right|\] are interchanged, which one of the following statements regarding the value of the determinant is CORRECT?
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